In this paper we provide upper bounds for the
Hausdorff dimension of the singular set of minima of general
variational integrals
\[
\int_{\Omega} F(x,v,Dv)\ dx\;,
\]
where $F$ is suitably convex with respect to $Dv $ and Hölder
continuous with respect to $(x,v)$. In particular, we prove that
the Hausdorff dimension of the singular set is always strictly
less than $n$, where $\Omega \subset R^n$.